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Integral equation methods for singular problems and application to the evaluation of Laplace eigenvalues

Eldar Akhmetgaliyev
California Institute of Technology


Boundary integral equation methods represent a powerful tool for the numerical solution of a variety of problems in acoustics, electromagnetics, fluid mechanics etc. Together with Oscar Bruno at Caltech, we developed a numerical method for the solution of scattering problems with mixed Dirichlet/Neumann boundary conditions. We investigated and obtained the exact form of the solution singularities -- which arise at transition points where Dirichlet and Neumann boundary conditions meet. These singularities are incorporated in the numerical approach and are resolved via Fourier Continuation technique. The resulting method exhibits spectral convergence. Additionally, jointly with Nilima Nigam at SFU, we applied the mixed boundary value solver to Laplace eigenvalue problems. The challenge is in eigenvalue search as a result of the properties of the objective function, which requires scanning through a range of frequencies. We introduce an improved search algorithm, that allows to locate the eigenvalues using standard root-finding methods. We also apply another integral equation method for domains with corners for a mode matching problem in electromagnetics. Jointly with Ahmed Akgiray at Caltech, we calculated TE and TM modes for domains of specific geometry. Those are further used for antenna design.

Tuesday, December 3, 2013
11:00AM AP&M 2402